Scientific investigation of a fractional model based on hybrid nanofluids with heat generation and porous medium: applications in the drilling process

This article investigates graphite-aluminum oxide hybrid nanoparticles in water-base fluid with the addition of heat generation in the presence of a porous medium. The problem is formulated in terms of momentum and energy equations with sufficient initial and boundary conditions. The solution is investigated by using the Laplace transform method. It is observed that the velocity of the drilling fluid is controlled by adding hybrid nanoparticles as compared to simple nanofluids. In a similar way, the temperature of the fluid is reduced. Also, the heat transfer rate is boosted up to 37.40741% by using hybrid nanofluid compared to regular nanofluid. Moreover, the heat transfer rate was increased up to 11.149% by using different shapes of nanoparticles in the base fluid water. It is also observed that by using hybrid nanofluid skin fraction is boosted up at y = 0 and boosted down at y = 1.

www.nature.com/scientificreports/ several reasons, the solutions obtained here are significant. These solutions can be used to verify the precision of their results by experimentalists in various water filtration process applications. Engineers can utilize these findings to decrease the number of experiments and time spent on practical work by altering different factors during the drilling process. Al 2 O 3 was chosen for its capacity to improve drilling mud rheological, electrical, and thermal characteristics. Because of their shape, graphite ( Gr ) is used to offer reduced filtrate loss. Furthermore, the Al 2 O 3 , Gr combination is non-hazardous and ecologically safe, making it ideal for usage in sensitive settings and applications, the drilling process is represented in Fig. 1.

Model formulation
Let us suppose an incompressible Newtonian viscous hybrid nanofluid between two infinite parallel plates. Water is taken as a based fluid with the combination of graphite and aluminium oxide nanoparticles. The distance between plates is d in a coordinate system. It is assumed that both plates are at rest and the temperature of the left plate is 0 , while the right plate temperature is w . The flow of the fluid is due to free convection as shown in Fig. 2.   www.nature.com/scientificreports/ The Reynolds number is neglected, and no pressure gradient is assumed in the direction of flow. Maxwell's relations for the magnetic field and Newton's second law for velocity. Maxwell's set of equations is as follow 33 : where Moreover, the electromagnetic force is described as 33 where i is the unit vector in the x-direction. And − → V = u y, t i, 0, 0 By incorporating − → F em in momentum equation of unsteady incompressible we get the following set of partial differential equation 34 : where The energy equation is where σ hnf , ρC p hnf and K hnf are the heat capacity and thermal conductivity of nanofluids defined as 35 : The physical IBC are: the subsequent dimensionless variables are introduced for non-dimensionalization: (6) u y, 0 = 0, y, 0 = 0 for all y ≥ 0,

Results and discussion
The problem of hybrid nanofluid is solved by using the Laplace transform method. Results are computed and discussed for several embedded parameters. Results are based on the nanoparticle's thermophysical properties in Table 1. Results plotted in Fig. 4 reflects the effect of fractional parameter α on velocity and temperature distributions. Different graphs for different values of α obtained for velocity and temperature profiles. These graphs are obtained for various values of α while keeping other parameters constants. In addition, each graph is made for a distinct value of α , which represents a solution so in this manner we have obtained various solutions. For the Re K j u y, α j t    www.nature.com/scientificreports/ experimentalists, it is easy to select the best curve fitting graph, which can be in good agreement with the results of real data. When φ is increased from 0.01 to 0.04, the velocity of the hybrid nanofluid is decreased for both particles, φ s1 for one particle and φ s2 for the second one. Physically this phenomenon says that with increasing values of volume fraction, the viscous forces become stronger due to which velocity retards. This physical trend is noted in Fig. 5 for both nanoparticles volume fractions. Which makes a strong agreement with the physics of the volume fraction of hybrid nanofluids. The comparison of Al 2 O 3 , Gr (hybrid nanofluid) is shown with respect to water-base Gr and Al 2 O 3 nanofluid in Fig. 6. It is observed that the velocity of the Al 2 O 3 , Gr (hybrid nanofluid) is lower as compared to other nanofluids. Physically, this is true because the density of Al 2 O 3 , Gr (hybrid nanofluid) is greater than the density of other nanofluids which is a factor to decrease the velocity. Moreover, the temperature distribution shows the same behavior as that of the velocity profile. As Al 2 O 3 , Gr (hybrid nanofluid) has higher thermal conductivity than other nanofluids but less density. Consequently, Al 2 O 3 , Gr (hybrid nanofluid)   www.nature.com/scientificreports/ transfers more heat compared to other nanofluids and are less dense, as a result, the temperature distributions of Al 2 O 3 , Gr (hybrid nanofluid) is lower than other nanofluids. The effect of the shape factor on the velocity and temperature profiles has been depicted in Fig. 7. The hydrogen bonding of hybrid nanofluid produces an important argument in thermal conductivity thus the velocity and temperature distribution are enhanced. The temperature profile for blade shape hybrid nanoparticles is maximum while for brick shape nanoparticles the temperature distribution is minimum. Similarly, the velocity profile for blade shape is maximum and minimum for brick shape nanoparticles. Figure 8 illustrates the influence of heat generation in temperature and velocity field, it is notified that heat generation boosts the temperature and velocity. When Q is set to a higher value, it   www.nature.com/scientificreports/ indicates that the device has absorbed more heat, resulting in a weaker intermolecular attractive force, which causes the temperature and velocity profiles to rise. Figure 9 reflects the influence of Gr 1 on the velocity of hybrid nanofluid. Gr 1 is the ratio of thermal to viscous forces, the dominancy of thermal forces for the fluid velocity for enhancement? That is why the increasing behavior of velocity of hybrid nanofluid is pictured in Fig. 9. Figure 10 stipulates that enhancing the magnetic parameter M provides a considerable hindrance to the flow behavior. This hindrance is due to the fact that enlargement of M strengthens the Lorentz forces, which by nature are resistive forces. Therefore, due to these strong resistive forces' retardation occurs in the fluid flow, and a decrease in boundary layer thickness is seen. The last Fig. 11 illustrates the effect of the porosity parameter k 1 on velocity profile, which enhances the velocity of the fluid. This is due to a decrease in the resistance of the porous medium, which leads to an increase in the thickness of the momentum boundary layer. By putting α = 0.5, M = 1.5, Gr = 5, k 1 = 15, t = 1, Q = 0 and φ s1 = φ s2 = 0 our solution is reduced to the solution of Saqib et al. 40 , which is presented in Fig. 12. Table 2 highlights the variation of Nusselt number for classical, unitary, and hybrid nanofluids nanoparticles, comparative analysis with nanofluid and shape effect of nanoparticles. It is observed that the enhancement of the rate of heat transfer is boosting up to 37.40741% by using hybrid nanofluid compared to unitary nanofluids, and up to 11.14983% by adding differently shaped nanoparticles in the base fluids. The variation of skin fraction for both plates is present in Table 3. It is observed from this table that increasing the volume fraction of nanofluid increases the skin faction on a left plate while the opposite behavior on the right plate. It is also clear that blades shaped nanoparticles have a high skin fraction on a left plate while Brick's shape nanoparticles have a high skin fraction on the right plate.

Summery
In the water filtration process application, this mathematical model is developed and analyzed with the combination of nanoparticles which make hybrid nanofluid. The results (graphical and tabulated) are purely obtained by numerical inversion of the Laplace transform. The different parameters are analyzed for the velocity and temperature profile. The comparison is also explored for nanofluid and hybrid nanofluid with graphite and aluminium oxide nanoparticles. The main outcomes are as follows.  www.nature.com/scientificreports/ • The fractional parameter provides the range of solution for temperature and velocity profile which makes the analysis more general as compared to the classical one. • The hybrid nanofluid has enhanced the temperature and retard the velocity, which helps in the rate of heat transfer during the drilling process. • The shape effect of nanoparticles reduced the temperature and velocity profile, which means that different shapes of nanoparticles should be used to control the temperature and velocity of the hybrid nanofluid during the filtration process by adding different shapes of the nanoparticle. • Dissolving the hybrid nanocomposite ( Al 2 O 3 , Gr ) in a water-base enhances the rate of heat transfer up to 37.40741%, which is higher than unitary nanofluid. • The blade's shape of nanoparticles enhances the rate of heat transfer up to 11.14983% rather than other shapes of nanoparticles during the filtration process. • The skin fraction on both plates can be controlled by adding hybrid nanofluid, the volume fraction of nanoparticles, and by adding different shapes of nanoparticles.
Lastly, I endorse some suggestions for future research to the readers as below.
• The following concept can be applied to various geometries of calendrical coordinates.
• One way to explore the same concept for different types of fluids with slip boundary conditions. • Furthermore, some new fractional derivatives can be used to more effectively summarise present fluid models along with concertation equations.

Data availability
Data of this study will be made available from the corresponding author on reasonable request.